The search for subsurface hydrocarbon deposits typically involves a multifaceted sequence of data acquisition, analysis, and interpretation procedures. The data acquisition phase involves use of an energy source to generate signals that propagate into the earth and reflect from various subsurface geologic structures. The reflected signals are recorded by a multitude of receivers on or near the surface of the earth, or in an overlying body of water. The received signals, which are often referred to as seismic traces, consist of amplitudes of acoustic energy that vary as a function of time, receiver position, and source position and, most importantly, vary as a function of the physical properties of the structures from which the signals reflect. The data analyst uses these traces along with a geophysical model to develop an image of the subsurface geologic structures. An important aspect of the geophysical modeling is the availability of estimates of the velocity of propagation in the earth formation.
Wireline measurements are commonly made in a borehole for measurement of the velocities of seismic waves in earth formation. Many of these seismic wave-types are dispersive in nature, i.e., they have a velocity that is dependent upon frequency. The dispersive waves may include compressional waves, shear waves and Stoneley waves and may be generated by monopole, dipole or quadrupole sources. FIGS. 2A and 2B illustrate an example of dispersive flexural waves in an earth formation. The curve 201 shows the slowness (ordinate, in μs/ft) as a function of frequency (abscissa, in Hz). FIG. 2B shows simulated waveforms 211 when a medium that has the dispersive characteristics of 201 in FIG. 2A is excited by a source that has the frequency dependence shown by 203. As would be known to those versed in the art, the simulated waveforms depend on both the dispersion curve and the source spectrum. The abscissa in FIG. 2B is time (in ms) and the successive traces are at increasing distances from the source.
The so-called slowness-time coherence (STC) processing has been used for analysis of such dispersive waves. See, for example, Kimball. The STC is commonly defined over a 2-D grid of slowness S, (reciprocal of velocity) and a window starting time T. This 2-D grid is called the slowness-time (ST) plane.
                              ρ          ⁡                      (                          S              ,              T                        )                          =                              1            m                    ⁢                                                                      ∫                  T                                      T                    +                                          T                      W                                                                      ⁢                                                                                                                          ∑                                                                              s                            i                                                    ⁡                                                      [                                                          t                              +                                                                                                S                                  ⁡                                                                      (                                                                          i                                      -                                      1                                                                        )                                                                                                  ⁢                                δ                                                                                      ]                                                                                                                                      2                                    ⁢                                      ⅆ                    t                                                                              ∑                                                      ∫                    T                                          T                      +                                              T                        W                                                                              ⁢                                                                                                                                                                  s                            i                                                    ⁡                                                      [                                                          t                              +                                                                                                S                                  ⁡                                                                      (                                                                          i                                      -                                      1                                                                        )                                                                                                  ⁢                                δ                                                                                      ]                                                                                                                      2                                        ⁢                                          ⅆ                      t                                                                                            .                                              (        1        )            In eqn. (1), ρ is the slowness, si is the signal, and δ is the quantization of the slowness grid. FIG. 3 shows an example of the STC plot for the example of FIGS. 2A-2B. The abscissa of FIG. 3 is the arrival time T (in ms) while the ordinate is the slowness in μs/ft. The nature of the STC display of FIG. 3 brings out the problem: how can the formation velocity (in this case, the formation shear velocity) be determined from the STC plot?
The present disclosure addresses the problem of estimation of formation slowness for a dispersive wave propagating in the earth formation.